How to do the sum over integer compositions

Question

How to do summation or how to find another representation of the sum that runs over integer compositions. $$ \sum_{r_1+ \ldots + r_{L}= k} \left(-\frac{ a}{1+a} \right)^{k-L} n^{(3 r_1)} (n+3 r_{1}-1)^{(3 r_{2})}\cdot \ldots \cdot (n+\sum_{i=1}^{L-1} 3 r_i- L+1)^{(3 r_{L})} $$ where $\sum r_i = k$ runs over all integer compositions of $k$ and $L$ is a number of parts in the composition and $n^{(3 r_1)}$ is the rising factorial. $n$ is a positive integer or zero. I mean usual definition of compositions as in https://en.wikipedia.org/wiki/Composition_(combinatorics).

Realization in Mathematica may become somehow useful

Sum[(-(a/(1 + a)))^(p - r) Product[Pochhammer[n + Plus@@Table[3 k[[i]]-1, {i, 1, j - 1}],3 k[[j]]],{j, 1, r}],{r, 1, p},{k,Compositions[p-r,r]+1}]

Answer 1

Let $n$ be fixed.

The sum in question can rewritten as $$S_k:=\frac{1}{(n-1)!}\sum_{L=1}^k\sum_{r_1+\dots+r_L=k} (n+3k-L)!\cdot \alpha^{k-L}\cdot f(n,k,L),$$ where $\alpha:=-\frac{a}{a+1}$ and $$f(n,k,L) := \sum_{0<s_1<\dots<s_{L-1}<k}\ \prod_{i=1}^{L-1} (n+3s_i-i).$$ (think of $s_i = r_1+\dots+r_i$)

This function for $L>1$ satisfies the recurrence: $$f(n,k,L) = \sum_{t=1}^{k-1} f(n,t,L-1)\cdot (n+3t-L+1).$$ For the generating function $$F(x,y) := \sum_{k=1}^\infty \sum_{L=1}^k f(n,k,L) x^k y^{n+3k-L}$$ it implies a linear differential equation: $$F(x,y) = \frac{xy^3}{1-xy^3}F'_y(x,y)+\frac{y^{n+2}x}{1-xy^3}$$ with a known solution.

Going back to the original sum, we apply Laplace transform to derive:

$$S_k = \frac{1}{(n-1)!\alpha^n}\ [x^k] \int_0^\infty F\big(\frac{x}{\alpha^2}, \alpha t\big) e^{-t} {\rm d}t,$$ where $[x^k]$ is the operator taking the coefficient of $x^k$.

I did not try much to simplify the result, but it gives a closed form expression for (the generating function of) $S_k$ nevertheless.